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I am really confused about the difference between $T_1$ space and $T_2$ space. To me they both seem to both have the same definition.

If $X=\{a,b,c,d\}$, what are the Topologies on $X$ that are $T_1$ but not $T_2$ and why?

MrDi
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1 Answers1

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There's no difference between $T_1$ and $T_2$ when considering finite topological spaces.

The reason is that a finite topological space has only finitely-many open sets. $T_1$ implies that a point $\{x\}$ is the intersection of all the open sets containing $x$. But since there are only finitely-many open sets, and finite intersections of open sets are open, it follows that $\{x\}$ is open. But then this means that the topology is discrete, in particular is $T_2$ (and much more).

This means we need to go to topological spaces with infinite underlying sets to find examples of spaces which are $T_1$ but not $T_2$. Such an example is the cofinite topology on any infinite set, say $\mathbb{N}$.

Hayden
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  • Does that mean that for any $X={a,b,c,...}$ where $X$ is finite then $T_{1}$ space is a $T_{2}$ space? – MrDi Dec 15 '16 at 03:23
  • Yes, precisely. For finite $X$, $T_1$ is the same as $T_2,T_3,T_4,T_5,T_6$ (since it is discrete and so a metrizable) – Hayden Dec 15 '16 at 03:23